There are several stations operated by JR and named after the cities' names but not the main station of the cities.
One of them is Yokkaichi station in Yokkaichi, Mie prefecture. Yokkaichi is the most populous city in Mie which is an important industrial and harbor city relatively close to Nagoya. Yokkaichi station serves for JR Kansai main line which connects Nagoya directly.
However, citizens of Yokkaichi mostly take Kintetsu trains to Nagoya from Kintetsu Yokkaichi station. The daily ridership of Kintetsu Yokkaichi is ten times bigger than the JR's one. The reason is very simple. Kintetsu Yokkaichi is much closer to city center and more well developed. Yokkaichi station was built mainly for freight trains which carries from and out from industrial zone which is located at the bay of Ise. Citizens of Yokkaichi call Kintetsu Yokkaichi station as "Yokkaichi station" and Yokkaichi station as "JR Yokkaichi station". In case you need to go to Yokkaichi and need a pick up there or meet someone at the station, please mention JR Yokkaichi station otherwise your driver or your friend will be waiting at Kintetsu Yokkaichi.
If you are Japan Rail Pass holder or traveling from Tokyo and wants to travel to Ise jingu shrine in Ise city, you should take rapid train Mie from Nagoya to Ise-shi and you will pass Yokkaichi station. On the other hand, if you are traveling from Osaka, Nara or Kyoto, take Kintetsu train to Ise-shi.